Before I discuss differentiating trigonometric functions, we should probably review a few key useful limits.
0. Useful Statements. We will use some statements that may or may not be obvious.
1. Sinc Limit. We will consider a function (called the “sinc” function) . What happens as ? Lets think about it.
We will consider , then we know
Divide through by
Well, we know that
Thus we have our limit bounded
We rewrite this to stress the boundedness:
More importantly, this implies (taking the recipricol):
This is one important limit.
2. Cosine Version. This is all very nice, but lets consider the limit
What to do? Well, we can do the following trick
The numerator (top of the fraction) becomes
We plug this in, rewriting our limit as
What to do now?
We use the squeeze theorem, noting that the fraction has bounds
Using limit properties, we find that
Thus our problem has bounds
The squeeze theorem produces the solution
Rewriting the function in familiar notation, we have
3. Punchline. So, we have just proven two important limits:
Exercise 1. Prove or find a counter-example: for any fixed constant we have .